Chapter 1, Problem 36RE

(a)

To determine

To find: The domain of the function $y=-1+\sqrt[3]{2-x}$ .

Answer to Problem 36RE

The domain of the function $y=-1+\sqrt[3]{2-x}$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given information:

The given function is $y=-1+\sqrt[3]{2-x}$ .

Calculation:

The function $f\left(x\right)=x^{\frac{1}{3}}$ is defined for all the real values of $x$ .

Similarly, the function $g\left(x\right)=\sqrt[3]{2-x}$ is defined for all the real values of $x$ .

So, the function $y=-1+\sqrt[3]{2-x}$ is defined for all the real values of $x$ .

Therefore, the domain of the function $y=-1+\sqrt[3]{2-x}$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

(b)

To determine

To find: The range of the function $y=-1+\sqrt[3]{2-x}$ .

Answer to Problem 36RE

The range of the function $y=-1+\sqrt[3]{2-x}$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

Explanation of Solution

Given information:

The given function is $y=-1+\sqrt[3]{2-x}$ .

Calculation:

The range of the function $f\left(x\right)=x^{\frac{1}{3}}$ is the set of real numbers.

So, the range of the function $y=-1+\sqrt[3]{2-x}$ is the set of real numbers of the interval $\left(-\infty ,\infty \right)$ .

Therefore, the range of the function $y=-1+\sqrt[3]{2-x}$ is the set of real numbers or the interval $\left(-\infty ,\infty \right)$ .

(c)

To determine

To graph: The function $y=-1+\sqrt[3]{2-x}$ .

Explanation of Solution

Given information:

The given functionis $y=-1+\sqrt[3]{2-x}$ .

Graph:

To graph a function $y=-1+\sqrt[3]{2-x}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator, press $\boxed{\text{Y}=}$ key and enter right hand side of the equation $y=-1+\sqrt[3]{2-x}$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{(-)}\boxed{1}\boxed{+}\boxed{(}\boxed{2}\boxed{-}\boxed{\text{X}}\boxed{)}\boxed{^}\boxed{(}\boxed{1}\boxed{÷}\boxed{3}\boxed{)}$ .

The display will show the equation,

  ${\text{Y}}_{\text{1}}=-1+\left(2-\text{X}\right)^\left(1/3\right)$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of both the equations as shown below.

  

Figure (1)

Interpretation: From the above graph it can be observed that the graph the function $y=-1+\sqrt[3]{2-x}$ is the shifted 2 units downwards, 1 unit to the right and reflection across y- axis of its parent function $y=\sqrt[3]{x}$ .